The system is initially prepared at the fully polarized ground state \(\rho _0 = |{\downarrow }\rangle \langle {\downarrow }|\)  of a deep trivial Hamiltonian with \(|m_z|\gg |\xi _0|\) . After quenching \(m_z\)  to a nontrivial value at time \(t_{n}=0\) , the system starts to evolve under the post-quench Hamiltonian \(\mathcal {H}(\mathbf {k},t_{n})\) ; see Fig. REF a.
Without noise, the spin polarization \(\langle (\mathbf {k},t_{n})\rangle \equiv \text{Tr}[\prod ^{n-1}_{i=0}U(t_{n-i},t_{n-i-1})\rho _0\prod ^{n-1}_{i=0}U^{\dagger }(t_{i+1},t_{i})]\)  precesses with respect to the Hamiltonian vector \(\mathbf {h}\) ; see Fig. REF a. The post-quench QAH phase can be determined by the dynamical topology emerging on BISs [1]}, identified as the momentum subspaces with \(h_{z}=0\) , where the initial state is perpendicular to the SO field \(\mathbf {h}_{{\rm {so}}}\equiv (h_{x},h_{y})\) , leading to vanishing time-averaged spin polarizations.
